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4                         ANALYTICAL MECHANICS
Solving these we get
tan a
DISCUSSION.—The last expression gives the value of « for a given value of M- When /z = 1, a. = 0, therefore in this raw the ladder will be
in equilibrium at any angle between 0 and 4> with the1 ground.    Kvidently
this is true for any value of /* greater than unity.
3. Find the smallest force which, when applied at the center of a carriage wheel of radius a, will drag it over an obstacle,
The forces acting on the wheel are: its weight W, the required force F, and the reaction R. Since the first two moot at the* center of the wheel, the direction of R must pass through the center also. Take I he eoorditwf e axes along and at right angles to R, as shown in Fig. 3«">, and let F make an angle 0 with the z-axis. Then the equations of equilibrium become
IX m F COS $ - R + W COH « --  0,
SF mptinG— IF sin a = 0, Zffo's W • a sin a — F sin 0 «a =- 0.
Prom either of the last two equations we get
F* „ ?»i.....« w
sin 0
Since W and a are fixed F can be changed only by changing 0.   Therefore the minimum value of F is given by the maximum value of nia 0, i.e.,
0 = -, which makes